### Quick description

Suppose one wants to find the parameter that minimizes a quantity , where is an increasing non-negative function in and is a decreasing non-negative function in . One can of course use calculus methods to do this, by finding the value(s) of where the derivative of vanishes. But a quick and dirty way to find the minimum *approximately* is just find the value where the two functions agree:

Indeed, since for , and for , we see that the minimal value of lies between and .

More generally, once one finds a where and are comparable in magnitude, this is already enough to compute the minimum of up to multiplicative constants.

### Prerequisites

Calculus

### Example 1

(Optimize expressions such as )

### General discussion

When optimizing a sum , where the intermediate are somehow "in between" and , the above heuristic is often effective (up to a factor of , perhaps) if one looks for the which balances the two extreme terms and . (Need an example of this...)

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